Finite-difference frequency-domain method

The finite-difference frequency-domain (FDFD) method is a <a href="/facts/Numerical_solution/QT9cYa9z">numerical solution</a> method for problems usually in <a href="/facts/Electromagnetism/oAkmALHT">electromagnetism</a> and sometimes in <a href="/facts/Acoustics/2GpQUymy">acoustics</a>, based on <a href="/facts/Finite-difference_approximation/EoG81Qn9">finite-difference approximations</a> of the <a href="/facts/Derivative_operator/Nr15J0IE">derivative operators</a> in the <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a> being solved.
While "FDFD" is a generic term describing all frequency-domain finite-difference methods, the title seems to mostly describe the method as applied to scattering problems. The method shares many similarities to the <a href="/facts/Finite-difference_time-domain/PvvfRVJo">finite-difference time-domain</a> (FDTD) method, so much so that the literature on FDTD can be directly applied. The method works by transforming Maxwell's equations (or other partial differential equation) for sources and fields at a constant frequency into matrix form 
 
 
 
 A
 x
 =
 b
 
 
 {\displaystyle Ax=b}
 
. The matrix A is derived from the wave equation operator, the column vector x contains the field components, and the column vector b describes the source. The method is capable of incorporating anisotropic materials, but off-diagonal components of the tensor require special treatment.
Strictly speaking, there are at least two categories of "frequency-domain" problems in electromagnetism. One is to find the response to a <a href="/facts/Current_density/IlUKSe2B">current density</a> J with a constant frequency ω, i.e. of the form 
 
 
 
 
 J
 
 (
 
 x
 
 )
 
 e
 
 i
 ω
 t
 
 
 
 
 {\displaystyle \mathbf {J} (\mathbf {x} )e^{i\omega t}}
 
, or a similar time-harmonic source. This frequency-domain response problem leads to an 
 
 
 
 A
 x
 =
 b
 
 
 {\displaystyle Ax=b}
 
 system of linear equations as described above. An early description of a frequency-domain response FDTD method to solve scattering problems was published by Christ and Hartnagel (1987). Another is to find the <a href="/facts/Normal_mode/XCsvQpuI">normal modes</a> of a structure (e.g. a waveguide) in the absence of sources: in this case the frequency ω is itself a variable, and one obtains an <a href="/facts/Eigenproblem/8TjEoT8u">eigenproblem</a> 
 
 
 
 A
 x
 =
 λ
 x
 
 
 {\displaystyle Ax=\lambda x}
 
 (usually, the eigenvalue λ is ω2). An early description of an FDTD method to solve electromagnetic eigenproblems was published by Albani and Bernardi (1974).

Finite-difference frequency-domain method open-in-new

Finite-difference frequency-domain method